CORRESPONDENCES IN ARAKELOV GEOMETRY AND APPLICATIONS TO THE CASE OF HECKE OPERATORS ON MODULAR CURVES
RICARDO MENARES
Abstract. In the context of arithmetic surfaces, Bost defined a generalized Arithmetic 2 Chow Group (ACG) using the Sobolev space L1. We study the behavior of these groups under pull-back and push-forward and we prove a projection formula. We use these results to define an action of the Hecke operators on the ACG of modular curves and to show that they are self-adjoint with respect to the arithmetic intersection product. The decomposition of the ACG in eigencomponents which follows allows us to define new numerical invariants, which are refined versions of the self-intersection of the dualizing sheaf. Using the Gross-Zagier formula and a calculation due independently to Bost and K¨uhnwe compute these invariants in terms of special values of L series. On the other hand, we obtain a proof of the fact that Hecke correspondences acting on the Jacobian of the modular curves are self-adjoint with respect to the N´eron-Tate height pairing.
Contents 1. Introduction1 2 2. Functoriality in L1 5 2 2.1. The space L1 and related spaces5 2 2.2. Pull-back and push-forward in the space L1 6 2 2.3. Pull-back and push-forward of L1-Green functions 12 3. Pull-back and push-forward in Arakelov theory 16 2 3.1. The L1-arithmetic Chow group, the admissible arithmetic Chow group and intersection theory 16 3.2. Functoriality with respect to pull-back and push-forward 18 4. Hecke correspondences on modular curves 21 4.1. Integral models of the modular curves X0(N) and Hecke correspondences 21 4.2. Arithmetic Chow group of X0(N) 23 4.3. Arithmetic Chow group of X0(N) with real coefficients 28 4.4. Self-intersection and refined self-intersection of the dualising sheaf 30 2 5. Appendix: cohomology of L1 spaces on compact Riemann surfaces 34 References 36 arXiv:0911.0546v3 [math.NT] 6 Mar 2013
1. Introduction Let X be an arithmetic surface, that is a projective, regular, 2-Krull dimensional scheme over Spec(Z). Such an object arises naturally as a model over the integers of a smooth algebraic curve defined over a number field. Arakelov devised a formalism on X analogous to the classical intersection theory on an algebraic surface defined over a field ([Ara74]). The basic object of this formalism is a compactified divisor, that is a pair (D, g), where D is a Weil divisor on X and g is a real-valued distribution on the compact Riemann surface X = X (C). Here, g is a Green function for D, subject to the following condition. 1 Suppose for simplicity that the genus of X is positive, and take an orthonormal basis ω1, . . . , ωn of the space of holomorphic 1-forms (where we take this space to be endowed i R with the inner product (α, β) 7→ 2 X α ∧ β). The canonical Arakelov 2-form is defined by n i X µ := ω ∧ ω . Ar n j j j=1 This is a C∞ volume form on X. It is independent of the choice of the orthonormal basis {ω1, . . . , ωn}. We impose that g satisfies the equation of currents
c (1.1) dd g + δD(C) = deg D(C) µAr. If this condition is satisfied, then g is said to be admissible. We also impose the normal- R ization X gµAr = 0. The key analytic ingredient needed to define an intersection number from two com- pactified divisors (D1, g1) and (D2, g2) is the star product
c (1.2) g1 ∗ g2 = g1(dd g2 + δD2(C)) + g2δD1(C).
This is a well defined current if D1(C) and D2(C) do not have points in common. The R integral g1 ∗ g2 is bilinear and symmetric in (g1, g2). X (C) In Gillet and Soul´e’ssubsequent treatment of Arakelov’s theory [GS90], the admissi- bility condition on g is dropped. More precisely, g is a real-valued C∞ function on the P complement of the support of D(C) = P aP [P ]. It is required that for every point P in this support, there is a local expansion of the form
2 (1.3) g = −aP log |θP (·)| + b, ∞ with θP a local chart vanishing at P and b a C function. Such a distribution is called a Green current for D. Equation (1.1) implies the local expansion (1.3), making Gillet and Soul´e’stheory a generalization of Arakelov’s theory. The star product (1.2) between normalized Green currents retains the symmetry and bilinearity properties. In this paper we study the lack of functoriality problem in Arakelov theory. That is, let ϕ : X → Y be a finite morphism between arithmetic surfaces, and (D, g) a compactified divisor on X . If the induced holomorphic covering ϕC : X (C) → Y(C) is ramified, then the pair (ϕ∗D, ϕ∗g) induced from the the push-forward is not a compactified divisor on Y, in either the Arakelov or the Gillet-Soul´esense. The reason is purely analytic in nature; ϕ∗µAr has singularities at the branching points, hence it is not the Arakelov canonical form on Y(C). Furthermore, ϕ∗g is not a Green current for ϕ∗D, because the push-forward operation does not preserve C∞ functions. In this sense, the theories of Arakelov and Gillet-Soul´eare not functorial. In many situations, it is natural to consider correspondences on arithmetic surfaces. Unfortunately, the interaction with the intersection formalism is excluded by this phenomenon. 2 Bost brought to this circle of ideas the Sobolev space L1, which consists of distributions in L2 whose derivatives also belong to L2. He proposed a generalization of Gillet-Soul´e’s theory for arithmetic surfaces ([Bos99]). In this framework, g is a real-valued element of 2 L1 away from the support of D(C) and for every point P in this support, there is a local 2 2 expansion similar to (1.3) but where b is in L1. Such an objet is called a L1-Green function. 2 The integral of the star product (1.2) between L1-Green functions is well defined. This 2 theory is optimal in the sense that L1 is the biggest space such that a useful (integral 2 of a) star product can be defined. Bost’s theory also improves on the geometrical part, because it doest not require the scheme X to be regular but only integral and normal. We drop the regularity in the definition of arithmetic surface in what follows. 1 2 Let us write CH[(X ), <, > to denote the L1 arithmetic Chow group defined in [Bos99]. In this paper, we prove that these groups have a good functoriality. More precisely, we provide a proof of the following Theorem 1.1. Let ϕ : X → Y be a generically finite morphism between integral, normal, arithmetic surfaces. (1) The natural definitions of pull-back and push-forward induce homomorphisms ϕ∗ : CH[1(Y) −→ CH[1(X )
1 1 ϕ∗ : CH[(X ) −→ CH[(Y). (2) Projection formula: for all x ∈ CH[1(X ) and y ∈ CH[1(Y), we have that ∗ <ϕ y, x> =
1 holds in the sense of currents. We denote by CH[(X )µ the corresponding µ-admissible arithmetic Chow group. We show that pull-back and push-forward are well-behaved with respect to this notion, as well as study how the reference 2-current changes. 2 2 Theorem 1.2. Let µ ∈ L−1 X (C) and ν ∈ L−1 Y(C) be real-valued 2-currents such that R µ = R ν = 1. The natural definitions of pull-back and push-forward induce X (C) Y(C) homomorphisms
∗ 1 1 ϕ : CH[(Y)ν −→ CH[(X ) ϕ∗ν deg ϕ 1 1 ϕ∗ : CH[(X )µ −→ CH[(Y)ϕ∗µ. Here, the main issue is the question of the existence of such µ-admissible Green func- tions (cf. Proposition 3.2). Such a distibution always exists locally, a result coming from the theory of elliptic PDEs. This reduces our problem to patching together local solu- 2 tions to obtain a global one, which in turn is an exercise in L1-cohomology of Riemann surfaces. This exercise is probably well known to the experts, though for lack of suitable 3 reference in the literature, we collect in the appendix the necessary facts and deductions to complete it. We mention that Burgos, Kramer and K¨uhn[BGKK07] have developed an arithmetic intersection theory which, in the case of arithmetic surfaces considered here, lies strictly in between Gillet-Soul´e’sand Bost’s. They show that their theory is functorial. This theory has the advantage of being able to handle higher dimensional arithmetic varieties, where no optimal theory analogous to Bost’s is known (though there are partial results in this direction due to A. Moriwaki [Mor98]). As an application, which is also our main motivation to study functoriality in this con- ∗ text, we consider the case of the modular curves X0(N) = Γ0(N)\H and their Hecke cor- respondences. We use the integral model X0(N) obtained by the modular interpretation in terms of Γ0(N)-structures as in [KM85]. On the other hand, the hyperbolic measure on H descends on X0(N) to a finite measure. We denote by µN the normalized 2-current 2 1 on X0(N). This is an element of L−1 X0(N) . We define CHd(N) := CH[(X0(N))µN .
Theorem 1.3. The Hecke correspondences Tl with l - N and the involutions wd with ˆ 1 d|N induce homomorphisms Tl and wˆd on CH[(X0(N)). These homomorphism form a ˆ commutative algebra, and the induced operator Tl is self-adjoint. Moreover, this algebra preserves CHd(N). We remark that, as the singularities of an admissible Green function with respect to µN are of log log type, the part of this statement concerning functoriality can also be obtained via the Burgos-Kramer-K¨uhntheory. num From now on, we suppose that N is squarefree. We denote by CHd R (N) the variant of the arithmetic Chow group CHd(N) obtained by taking divisors (compactified, µN - admissible) with real coefficients, and then the quotient by the numerical equivalence relation (cf. Section 4.3 for a precise definition). This is a real vector space of finite ˆ dimension, and the Hecke operators Tl andw ˆd act on it. The Hodge index theorem in this num context allows us to decompose CHd R (N) as a sum of two Hecke-invariant subspaces:
num ⊥ (1.4) CHd R (N) = Eis ⊕ J, where Eis is the space spanned by compactified irreducible components of fibers, and
∼ (1.5) J = J0(N)(Q) ⊗ R as real vector spaces. It is possible to choose the isomorphism in (1.5) such that the Hecke actions on both sides are compatible (Corollary 4.11). Hence we may further decompose
M (1.6) J = Jf , f where f runs through a basis of S2 Γ0(N) consisting of eigenforms. Let ω be the dualising sheaf on X0(N). By taking the divisor induced by a section of 2 this sheaf with an appropriate choice of L1-Green function (cf. Section 4.4), we obtain a classω ˆ ∈ CHd(N). Following the decompositions (1.4) and (1.6) we can then write
X ωˆ =ω ˆEis +ω ˆJ , ωˆJ = ωˆf . f 4 We consider the self-intersection of each component as a refined invariant:
2 2 2 2 X 2 ωˆ =ω ˆEis +ω ˆJ , ωˆJ = ωˆf . f Using a calculation due to U. K¨uhn([K¨uh01]), we then obtain
Theorem 1.4. Suppose that N is squarefree. Let gM := genus of X0(M). We have that
2 ωˆEis 576 1 0 X log p (1.7) 2 = ζ(−1) + ζ (−1) − , (gN − 1) [Γ0(1) : Γ0(N)] 2 gN − 2g N + 1 p|N p where ζ(·) is the Riemann zeta function.
One of the divisors in the class ωJ is an explicit linear combination of Heegner points of discriminants -3 and -4. Let Hi (resp. Hj) be the sum of all divisors on X0(N) of the 1 1 form 2 [P ]−[∞] (resp. 3 [P ]−[∞] ), with P a Heegener point of discriminant -4 (resp. -3). Using the Gross-Zagier formula, we obtain that Theorem 1.5. Suppose that f is a normalized newform, and that N is a squarefree integer such that (6,N) = 1. Then,
2 2 1/2 1/2 (1.8)ω ˆf = − hNT (Hi)f + 2hNT (Hj)f .
where hNT is the N´eron-Tate height on J0(N)(Q). Furthermore, if N is prime, then √ 1 3 h (H ) = L(f, χ , 1)L0(f, 1), h (H ) = L(f, χ , 1)L0(f, 1), NT i f 2π2(f, f) −4 NT j f 4π2(f, f) −3 where√ χ−4 (resp. χ−3) is the quadratic Dirichlet character associated to Q(i)/Q (resp. Q( −3)/Q) and (f, f) is the Petersson norm of f.
Though we have treated in detail only the case of Γ0(N), an analogous analysis can be made for Hecke operators on other congruence groups, as well as on Shimura curves. Acknowledgements: This work is based in part on the author’s Ph.D. thesis at Uni- versit´ed’Orsay, written under the supervision of E. Ullmo. It was A. Chambert-Loir who suggested viewing the results of the thesis as a functoriality problem on an appropiate function space. We have benefited from useful concrete suggestions by J.-B. Bost, U. K¨uhnand A. Thuillier.
2 2. Functoriality in L1 2 2.1. The space L1 and related spaces. We recall the basic definitions from [Bos99]. 2 Let X be a compact Riemann surface. The space L1(X) is the space of distributions 2 ∂ ∂ 2 f ∈ L (X) such that ∂z f and ∂z¯f are also in L (X). We have a hermitian inner product 2 (·, ·)1,X on L1(X) given by Z (f, g)1,X = i ∂f ∧ ∂g. X Its associated seminorm is given by Z 2 kfk1,X = i ∂f ∧ ∂f ∈ R≥0. X 5 Let ν be a 2-form on X. We consider formally the pairing Z (2.1) (f, g) 7→ fgν¯ + (f, g)1,X . X If ν is positive and continuous, then the pairing (2.1) is a non degenerate, hermitian inner 2 product on L1(X), endowing it with a Hilbert space structure. We denote by k · kν the 2 associated norm. As X is compact, the topology on L1(X) induced by this norm does not ∞ 2 depend on the choice of ν. The subspace C (X) is dense in L1(X) under this topology. 2 We can also make L1 into a sheaf. For an open set U ⊆ X, we denote by νU the 2 restriction to U of the given continuous volume form. We define L1(U) as the space of 2 ¯ 2 distributions on U which are L with respect to νU and such that ∂f, ∂f are L . Again, 2 L1(U) is independent of the choice of ν, as well as the choice of ambient Riemann surface, cf [Bos99], p.255. 2 We define L−1(X) to be the space of 2-currents µ that can be locally written on open c 2 sets U as µ = dd f with f ∈ L1(U). 2 Remark 2.1. This definition of L−1 is equivalent to the one given in the introduction (cf. Theorem 5.1 in the appendix). Let D(U) denote the space of C∞, complex valued functions on X whose support is 2 a compact set contained in U. We define L1(U)0 to be the closure of the image of the 2 2 2 natural map D(U) → L1(X). As X is compact, we have that L1(X)0 = L1(X). Similarly, we define L1(U) as the space of distributions on U which are L1 with respect 1 1 to νU , with L (U)0 the closure of the image of the natural map D(U) → L (X). We define formally Z (f, g)1,U := i ∂f ∧ ∂g. U Let us record for latter use the following result. Lemma 2.2. Assume either of the following hypotheses: 2 2 • f ∈ L1(U)0, g ∈ L1(U) 1 ∂ 1 ∞ • f ∈ L (U)0, ∂z f ∈ L (U), g ∈ C (U). Then, we have an equality of absolutely convergent integrals:
Z c (2.2) 2π fdd g¯ = −(f, g)1,U . U Proof: It suffices to consider f ∈ D(U). Then, the identity d(f∂g) = ∂f ∧∂g −2πifddcg¯ and the vanishing of f on ∂U prove (2.2). Since either of the hypotheses ensure that (f, g)1,U is absolutely convergent, this finishes the proof.
2 2.2. Pull-back and push-forward in the space L1. Let ϕ : X → Y be a non-constant holomorphic map between compact Riemann surfaces. For a function g ∈ C∞(X) (resp. ∞ ∗ ∗ f ∈ C (Y )), the functions ϕ f and ϕ∗g are given by ϕ f = f ◦ ϕ and
X (2.3) ϕ∗g(y) = exg(x), x∈ϕ−1(y) ∗ ∞ where ex is the ramification index of ϕ at x. We have that ϕ f ∈ C (X). Although ϕ∗g is continuous, it is not C∞ at branched points. However, we have the following result for the latter function. 6 ∞ Lemma 2.3. Let U ⊂ X be an open set. For any g ∈ C (U) the function ϕ∗g given by 2 (2.3) belongs to L1 ϕ(U) .
2 Proof: As ϕ∗g is continuous, it suffices to check that the derivative of ϕ∗g is L around the finite set of branched points contained in ϕ(U). Using local coordinates, we reduce the problem further to the following situation: g is a C∞ function on D = {z ∈ C, |z| < 1}, and ϕ : D → D is the map ϕ(z) = zn. Fixing a branch of the logarithm on D, we then have that X 1/n ϕ∗g(z) = g(ρz ) almost everywhere. ρn=1 Then,
(1/n)−1 ∂ϕ∗g z X ∂g (z) = h(z1/n) almost eeverywhere, where h(z) := ρ (ρz). ∂z n ∂z ρn=1 Hence,
Z ∂ϕ g 2 i Z i ∗ (z) dz ∧ dz¯ = |z|(2/n)−2|h(z1/n)|2dz ∧ dz¯ 2 D ∂z n D Z 2π/n Z 1 = 2 r|h(reiθ)|2drdθ, 0 0 as can be seen by an elementary change of variables. The last integral converges because h is smooth.
Proposition 2.4. The linear maps
∗ ∞ ∞ ∞ 2 ϕ : C (Y ) → C (X), ϕ∗ : C (X) → L1(Y ) 2 are continuous in the L1 topology. We postpone the proof of this statement to the end of this Section. ∞ ∞ 2 As the spaces C (X),C (Y ) are dense in the respective L1 topologies, Proposition 2 2.4 allows us to extend the pull-back and push-forward to the respective L1 spaces,
∗ 2 2 2 2 ϕ : L1(Y ) → L1(X), ϕ∗ : L1(X) → L1(Y ). We also have well-defined maps
∗ 2 2 2 2 ϕ : L−1(Y ) → L−1(X), ϕ∗ : L−1(X) → L−1(Y ), ∗ c c ∗ c c given locally by ϕ dd f := dd ϕ f and ϕ∗dd g := dd ϕ∗g.
Lemma 2.5. Fix an open set U ⊂ X. Assume either of the following hypotheses: 2 2 • f ∈ L1 ϕ(U) , g ∈ L1(U) ∂ ∗ 1 ∂ 1 ∞ • ∂z ϕ f ∈ L (U), ∂z f ∈ L ϕ(U) , g ∈ C (U). Then, we have that ∗ (ϕ f, g)1,U = (f, ϕ∗g)1,ϕ(U). 7 Proof: Either of the hypotheses above allows us to reduce to the case where f and g are C∞ (by a well known density argument). Then, using a partition of unity, the problem is further reduced to the case where ϕ is the covering z 7→ zn of the complex unit disc. In this situation, the asserted equality boils down to an elementary change of variables.
For a 2-current µ and a distribution f, we define formally the symbol
Z (2.4) f · µ := fµ. X Lemma 2.6. The integral in (2.4) is absolutely convergent under either of the following hypotheses: 2 2 (i) f ∈ L1(X), µ ∈ L−1(X) 1 ∂ 1 ∞ (ii) f ∈ L (X), ∂z f ∈ L (X), µ a C 2-form. Moreover, we have that ∗ 2 2 (1) ϕ s · µ = s · ϕ∗µ, for all s ∈ L1(Y ) and µ ∈ L−1(X) ∗ 2 2 (2) ϕ∗t · ν = t · ϕ ν, for all t ∈ L1(X) and ν ∈ L−1(Y ). Proof: Suppose hypothesis (i) holds. Using a partition of unity and the compactness of X, the integral in (2.4) splits as a finite sum of integrals over open sets U such that c 2 2 µ = dd h, with h ∈ L1(U) and f ∈ L1(U)0. Then, every integral on the sum converges by Lemma 2.2, which shows that (2.4) is absolutely convergent. The same argument works ∞ 1 under hypothesis (ii), taking h ∈ C (U) and f ∈ L (U)0. To prove (1), we choose a finite collection of open sets U as in the paragraph above. ∗ 2 As ϕ is a proper map, we may suppose that ϕ s ∈ L1(U)0. Using Lemma 2.2 and Lemma 2.5, we obtain that
Z ∗ c ∗ ¯ 2π α sdd h = −(α s, h)1,U U ¯ = −(s, α∗h)1,ϕ(U) Z c (2.5) = 2π sdd α∗h. ϕ(U)
The identity (2) is proved similarly.
Now, we establish three lemmas leading to the proof of Proposition 2.4. The first one 2 ∗ will be used to control the L norm of ϕ (resp. ϕ∗) away from the ramification (resp. branched) points.
Lemma 2.7. Let µX (resp. µY ) be a nonnegative, continuous 2-form on X (resp. on Y ). Let ϕ : X → Y be a non constant holomorphic map between compact Riemann surfaces, and let f ∈ C∞(Y ), g ∈ C∞(X). Then, the following relations hold:
∗ p kϕ fkµX ≤ deg ϕkfkϕ∗µX
∗ kϕ∗gkµY ≤ deg ϕkgkϕ µY .
Remark 2.8. We stress that, as ϕ∗µX is not continuous if ϕ is ramified, this lemma does not immediately imply Proposition 2.4 (cf. Remark 2.12). 8 ∞ 2 Proof: The space of C functions is dense in L1. In particular, we can approximate in 2 ∞ the L1 sense continuous functions by C functions. Hence, we may suppose without loss 2 2 of generality that µX ∈ L−1(X) and µY ∈ L−1(Y ). We use Lemma 2.6 and Lemma 2.5 to find that
kϕ∗fk2 = |ϕ∗f|2 · µ + (ϕ∗f, ϕ∗f) µX X 1,X ∗ 2 ∗ = ϕ |f| · µX + (f, ϕ∗ϕ f)1,Y 2 = |f| · ϕ∗µX + deg ϕ(f, f)1,Y ≤ deg ϕkfk2 . ϕ∗µX This proves the first assertion. We will now deduce the second from the first:
kϕ gk2 = ϕ g · (ϕ g)µ + (ϕ g, ϕ g) ∗ µY ∗ ∗ Y ∗ ∗ 1,Y = ϕ∗g · (ϕ∗g)µY + (ϕ∗g, ϕ∗g)1,Y ∗ ∗ = g · ϕ (ϕ∗g)µY + (g, ϕ ϕ∗g)1,X ∗ ∗ ∗ = g · ϕ ϕ∗g ϕ µY + (g, ϕ ϕ∗g)1,X . Applying the Cauchy-Schwarz inequality to the inner product ∗ (h, l) 7→ h · lϕ µY + (h, l)1,X , we obtain that
2 ∗ kϕ gk ≤ kgk ∗ kϕ ϕ gk ∗ ∗ µY ϕ µY ∗ ϕ µY p ∗ ∗ ≤ deg ϕkgkϕ µY kϕ∗gkϕ∗ϕ µY p ∗ = deg ϕkgkϕ µY kϕ∗gk(deg ϕ)µY
∗ ≤ deg ϕkgkϕ µY kϕ∗gkµY .
Lemma 2.3 ensures that kϕ∗gkµY is a real number. Hence, by the above inequality, we ∗ conclude that kϕ∗gkµY ≤ deg ϕkgkϕ µY .
Lemma 2.9. Let D := {|z| < 1} ⊂ C and let S = {f ∈ C∞(D): f = 0 on ∂D}. Then, for all f ∈ S, we have that Z ∂f 2 Z ∂f 2
dz ∧ dz¯ = dz ∧ dz.¯ D ∂z D ∂z¯ Proof: Using Lemma 2.2, we have that
Z ∂f 2 Z Z c ¯ (2.6) dz ∧ dz¯ = ∂f ∧ ∂f = 2πi fdd f. D ∂z D D Similarly,
Z ∂f 2 Z Z Z ¯ ¯ ¯ c dz ∧ dz¯ = ∂f ∧ ∂f = ∂f ∧ ∂f = 2πi fdd f. D ∂z¯ D D D The identity d(f∂f +f∂f¯ ) = 2πi(fdd¯ cf −fddcf¯), along with the vanishing on the bound- ary allow us to deduce the result.
9 The following lemma can be seen as a weighted Poincar´einequality. It will be used in the proof of Proposition 2.4 to control the L2 norm of ϕ∗ around the ramification points.
Lemma 2.10. Let S be the set defined in Lemma 2.9. For every δ > 0 there exists Cδ > 0 such that for all f ∈ S,
Z |f(z)|2 Z ∂f 2 (2.7) i dz ∧ dz¯ ≤ C i dz ∧ dz.¯ 2−δ δ D |z| D ∂z Proof: Observe that the integral on the left hand side is convergent. Indeed,
Z 2 Z 2π Z 1 2 |f(z)| |f| 4π 2 i 2−δ dz ∧ dz¯ = 2 1−δ drdθ ≤ sup |f| . D |z| 0 0 r δ D Let Dε = {ε < |z| < 1}. We have that
2 |z|δ |f|2 2 |z|δ ∂|f|2 (2.8) d |f|2dz¯ = dz ∧ dz¯ + dz ∧ dz¯ on . δ z¯ |z|2−δ δ z¯ ∂z Dε We remark that Z |z|δ Z 2π lim |f|2dz¯ = lim −iεδ |f(εeiθ)|2dθ = 0. ε→0 z¯ ε→0 ∂Dε 0 Then, using (2.8),
Z |f|2 Z |f|2 i dz ∧ dz¯ = lim i dz ∧ dz¯ |z|2−δ ε→0 |z|2−δ D Dε 2 Z |z|δ ∂|f|2 2 Z |z|δ = lim − i dz ∧ dz¯ − i |f|2dz¯ ε→0 δ z¯ ∂z δ z¯ Dε ∂Dε 2 Z |z|δ ∂|f|2 = − i dz ∧ dz.¯ δ D z¯ ∂z But
Z |z|δ ∂|f|2 Z ∂|f|2 δ−1 dz ∧ dz¯ ≤ i |z| dz ∧ dz¯ D z¯ ∂z D ∂z Z ∂f Z ∂f δ−1 δ−1 (2.9) ≤ i |z| f dz ∧ dz¯ + i |z| f dz ∧ dz,¯ D ∂z D ∂z¯ ∂ 2 ¯ ∂ ∂ where we have used the relation ∂z |f| = f ∂z f + f ∂z¯f. We bound the first integral using the Cauchy-Schwarz inequality. That is,
Z ∂f Z |f|2 1/2 Z ∂f 2 1/2 i |z|δ−1 f dz ∧ dz¯ ≤ i dz ∧ dz¯ i dz ∧ dz¯ 2−2δ D ∂z D |z| D ∂z Z |f|2 1/2 Z ∂f 2 1/2 ≤ i dz ∧ dz¯ i dz ∧ dz¯ . 2−δ D |z| D ∂z Using Lemma 2.9, the same bound applies to the second integral in (2.9). Putting everything together, we obtain that
Z |f|2 4 Z |f|2 1/2 Z ∂f 2 1/2 i dz ∧ dz¯ ≤ i dz ∧ dz¯ i dz ∧ dz¯ . 2−δ 2−δ D |z| δ D |z| D ∂z 10 2 This implies (2.7) with Cδ = (4/δ) .
Remark 2.11. Lemma 2.10 can be viewed as a variant of the Hardy inequality ([HLP88], Theorem 327). In both cases, the main idea is to express the weighted L2 norm of f as an integral involving the derivative of f. In our case, this is done through the equality (2.8).
Proof of Proposition 2.4 : the endomorphism of the unit disk D given by z 7→ zn pulls 2 2(n−1) ∗ back dz ∧ dz¯ to n |z| dz ∧ dz¯. Hence, the form ϕ µY on Lemma 2.7 is continuous, non negative, with zeroes at the ramification points. This proves the assertion about the push-forward in Proposition 2.4. Concerning the pull-back, we must prove that we can choose a positive, continuous 2-form µ = µ(ϕ, µX ) on Y as well as a constant C = C(ϕ, µX ) > 0, such that
∗ (2.10) kϕ fkµX ≤ Ckfkµ holds for all f ∈ C∞(Y ). ∗ ∗ Lemma 2.5 implies the equality (ϕ f, ϕ f)1,X = deg ϕ(f, f)1,Y , so it suffices to bound the L2 norm of ϕ∗f. Let E ⊂ Y be the set of branched points of ϕ. We select a convenient neighborhood U of E as follows: for each ramification point P ∈ X of ϕ with ramification index nP , there exists a neighborhood VP ⊂ X (resp. UP ⊂ Y ) of P (resp. of the branched point ϕ(P )) and local charts θP : D → VP , ηP : D → UP such that the following diagram commutes:
θP z / V _ D P ϕ
ηP nP z D / UP We may assume without loss of generality (shrinking the neighborhoods if required) that V P ∩ V Q = ∅ if P 6= Q. Let us put U := ∪P ramifiedUP . Let UE be another neighborhood of E such that U E ⊂ U. Let ψ1, ψ2 be a partition of unity subordinate to the covering {Y \U E,U}. Let
n ∂ ∂ o M := max sup |ψi|, sup ψi , sup ψi . i=1,2 i=1,2 ∂z i=1,2 ∂z¯ ∞ We split f ∈ C (Y ) as f = f1 + f2, where f1 := fψ1 and f2 := fψ2. As ∂f ∂f |f | ≤ M|f|, i ≤ M |f| + , kϕ∗fk ≤ kϕ∗f k + kϕ∗f k , i ∂z ∂z µX 1 µX 2 µX
it suffices to prove (2.10) for f1 and f2. Let µ be a positive continuous volume form such that µ = ϕ∗µX on Y \U E. Note that
such a form exists because ϕ∗µX is continuous on Y \E. We have that kf1kϕ∗µX = kf1kµ. Then, since suppf1 ⊂ Y \U E, it follows from Lemma 2.7 that
∗ p kϕ f1kµX ≤ deg ϕkf1kµ.
We may suppose that the support of f2 is contained in some UP . We then have that 11 Z Z ∗ 2 ∗ 2 |ϕ f2| µX = |ϕ f2| µX −1 X ϕ (UP ) Z X ∗ ∗ 2 ∗ = |θQϕ f2| θQµX ϕ(Q)=P D Z X ∗ ∗ 2 ≤ λ i |θQϕ f2| dz ∧ dz.¯ ϕ(Q)=P D The last inequality, valid for some λ > 0 depending only on the choice of charts, follows ∗ from the continuity of θQµX on D. Putting n := nP we then find that Z Z ∗ ∗ 2 n 2 i |θQϕ f2| dz ∧ dz¯ = i |f2 ◦ ηQ(z )| dz ∧ dz¯ D D Z 2 i |f2 ◦ ηQ(z)| = 2−(2/n) dz ∧ dz¯ n D |z| Z ≤ Cni ∂(f2 ◦ ηQ) ∧ ∂(f2 ◦ ηQ), D where the last follows by application of Lemma 2.10 to f2 ◦ ηQ ∈ D(D). Finally, Z Z ∂(f2 ◦ ηQ) ∧ ∂(f2 ◦ ηQ) = deg ϕ ∂f2 ∧ ∂f2 D UP Z = deg ϕ ∂f2 ∧ ∂f2. Y This achieves the proof of (2.10) for f2.
Remark 2.12. J.-B. Bost explained to us another way to handle the problem of the ∞ ∞ ramification points. Let Cϕ (Y ) be the subspace of C (Y ) consisting of functions whose support does not contain the set of branched points of ϕ. A result by Deny and Lions ∞ ([DL54], Th´eor`emeII.2.2), using Cartan’s theory of balayage, ensures that Cϕ (Y ) is dense 2 ∗ 2 2 in L1(Y ). Then to obtain a well defined pull-back map ϕ : L1(Y ) → L1(X), it suffices ∗ ∞ to show the continuity of ϕ on Cϕ (Y ), which can be easily done (e.g. using Lemma 2.7 and the classical Poincar´einequality). This approach avoids the use of Lemma 2.10, though is less elementary. 2.3. Pull-back and push-forward of L2-Green functions. Let X be a compact Rie- P 1 mann surface, and let D = P aP [P ] be a divisor with real coefficients on X. By definition, a Green function with C∞ regularity for D is a real-valued distribution h on c ∞ 1 X such that the current dd h + δD is a C 2-form . Remark 2.13. The above definition is equivalent to the following two properties: ∞ • h is C outside of the finite set |D| := {P such that aP 6= 0} • for every P ∈ |D|, there exists a local chart (U, θP ) centered at P such that
2 h = −aP log |θP (·)| + b,
1This is what Gillet and Soul´ecall a Green current. Following Bost, we use a different terminology better suited to the discussion of regularity issues. 12 on U, where b ∈ C∞(U). The first property is a consequence of the ellipticity of the ddc operator. The second one 2 c comes from the fact that log | · | is a fundamental solution of dd f = δ0 on the complex unit disc.
2 Definition 2.14. An L1-Green function for D is a real-valued distribution g on X such that there exists a Green function with C∞ regularity h for D, along with an element 2 ψ ∈ L1(X) such that
(2.11) g = h + ψ. We call this equality a regular splitting of g.
c Remark 2.15. A distribution g satisfies the above definition if and only if dd g + δD is 2 c a 2-current in L−1(X) (cf. Proposition 3.2 below). In particular, the 2-current dd g is a Radon measure, that is, it extends to a continuous functional on the space C0(X) of continuos functions (and not merely on C∞(X)). Let ϕ : X → Y be a non-constant holomorphic map between compact Riemann sur- faces. Let D be a divisor on X and let h be a Green function with C∞ regularity for D. We define the distribution ϕ∗h on Y by
X (2.12) ϕ∗h(y) := exh(x), x∈ϕ−1(y) where ex is the ramification index of ϕ at x. The definition above has to be understood in the following sense. Using Remark 2.13, we see that the right side of (2.12) defines a function on the complement of the finite set
{y ∈ Y such that ϕ−1(y) ∩ |D|= 6 ∅}.
It can be proven that the singularities of ϕ∗h on this set are logarithmic (cf. (2.15) below for a precise verification), thus defining a distribution on X. Finally, we define
ϕ∗g := ϕ∗h + ϕ∗ψ. Similarly, let E ∈Div(Y ) and let k be a Green function with C∞ regularity for E. We define the distribution ϕ∗k on X by
ϕ∗k := k ◦ ϕ.
This definition is made rigorous in the same way as we did it with ϕ∗h above. 2 Let f be a L1-Green function for E, with a regular splitting
(2.13) f = k + ξ, 2 where ξ ∈ L1(Y ). We define
ϕ∗f := ϕ∗k + ϕ∗ξ.
∗ 2 Theorem 2.16. The distribution ϕ∗g (resp. ϕ f) is a L1-Green function for ϕ∗D (resp. for ϕ∗E). 13 Proof: We first prove the assertion concerning the push-forward. It suffices to consider the case D = [P ]. Let n be the ramification index of ϕ at P , and let Q := ϕ(P ). We choose local charts (U, θP ) centered at P and (V, θQ) centered at Q such that the following diagram commutes:
θP (2.14) U / D ϕ α
θQ V / D. Here, α(z) = zn. We may assume that the ramification index is 1 for every point in U different from P , and that we have
2 h = − log |θP | + b on U, with b ∈ C∞(U). We may also assume that supp h ⊂ U 2. 2 As ϕ∗ψ ∈ L1(Y ), it suffices to prove that ϕ∗h can be expanded around Q as
2 ˜ ϕ∗h = − log |θQ(·)| + b, ˜ 2 where b ∈ L1 ϕ(U) . Using our assumptions on the support of h, we have that for all y ∈ V \{Q},
X ϕ∗h(y) = h(x)
x∈ϕ−1(y) x∈U
X 2 = − log |θP (x)| + b(x)
x∈ϕ−1(y) x∈U X 1 = − log |α ◦ θ (x)|2 + ϕ b(y) n P ∗ x∈ϕ−1(y) x∈U X 1 = − log |θ ◦ ϕ(x)|2 + ϕ b(y) n Q ∗ x∈ϕ−1(y) x∈U X 1 = − log |θ (y)|2 + ϕ b(y) n Q ∗ x∈ϕ−1(y) x∈U 2 = − log |θQ(y)| + ϕ∗b(y)(2.15) ˜ 2 Using Lemma 2.3, we have b := ϕ∗b ∈ L1 ϕ(U) . Now, we prove the assertion concerning the pull-back. As before, we may assume that E = [Q]. We have that
∗ X ϕ [Q] = eP [P ]. ϕ(P )=Q ∗ We fix P ∈ |ϕ [Q]| and choose local charts (U, θP ) centered at P and (V, θQ) centered at Q such that the diagram (2.14) commutes, with α(z) = zeP . Let us write f = k + ξ as in
2If this is not the case, we take a C∞ function s with support contained in U and such that s ≡ 1 in a neighborhood of P . Then, g = sh + ψ + (1 − s)h is also a regular splitting as in (2.11). 14 (2.13). This reduces the problem to a study of the expansion of ϕ∗k around P . We have that
ϕ∗k = k ◦ ϕ 2 = − log |θQ ◦ ϕ| + b ◦ ϕ eP 2 ∗ = − log |θP | + ϕ b 2 ∗ = −eP log |θP | + ϕ b. ∗ ∞ As ϕ b is C , this concludes the proof.
The following two Lemmas will be used in the proof of Theorem 3.5. We adopt the following notation. For a 2-current T and a test function φ ∈ C∞, we denote by T [φ] the value of T at φ. Lemma 2.17. Using the notations of Theorem 2.16, we have that
c ∗ c ∞ dd ϕ f[φ] = dd f[ϕ∗φ], φ ∈ C (X)(2.16) c c ∗ ∞ (2.17) dd ϕ∗g[φ] = dd g[ϕ φ], φ ∈ C (Y ).
Remark 2.18. As ϕ∗φ is continuous, the right hand side of (2.16) is well defined (cf. Remark 2.15). Proof: we write f = k + ξ. Using Lemma 2.6, we have that Z Z c ∗ ∗ c c c dd ϕ ξ[φ] = − ϕ ξdd φ = − ξdd (ϕ∗φ) = dd ξ[φ]. X Y c ∞ We put w := dd k + δE, which is a C 2-form. Using Lemma 2.6, we have that
c ∗ ∗ ∗ dd ϕ k[φ] = ϕ w[φ] − ϕ δE[φ] ∗ = ϕ w · φ − δE[ϕ∗φ]
= w · ϕ∗φ − δE[ϕ∗φ] c = dd k[ϕ∗φ], thus proving (2.16). Equation (2.17) is proved similarly.
The following lemma is a variant of Lemma 2.6 for Green functions. Lemma 2.19. Let h be a Green function with C∞ regularity for a divisor D on Y such that the support of ϕ∗h is contained in a neighborhood U of |ϕ∗D|. Let s ∈ C∞(U). Then Z Z ∗ c c ϕ hdd s = hdd ϕ∗s. U ϕ(U) ∗ ∂ ∗ 1 Proof: Theorem 2.16 ensures that ϕ h and ∂z ϕ are L on U (i.e., apply the local expansion in Remark 2.13 to ϕ∗h). Then, by Lemma 2.2 and Lemma 2.5, we have that
Z ∗ c ∗ 2π ϕ hdd s = −(ϕ h, s¯)1,U U = −(h, ϕ∗s¯)1,ϕ(U) Z c = 2π hdd ϕ∗s. ϕ(U) 15 3. Pull-back and push-forward in Arakelov theory 2 3.1. The L1-arithmetic Chow group, the admissible arithmetic Chow group and intersection theory. Let X be an arithmetic surface, that is, a projective, integral, normal scheme over Spec(Z) of Krull dimension 2. The ring OX (X ) is the ring of integers of a number field K. We denote by Div(X ) the set of Weil divisors on X . A compactified divisor (in the sense of Bost) is a pair (D, g), where D ∈Div(X ) and g 2 is a L1-Green function for the divisor D(C) on the compact Riemann surface X (C). In addition, g must be invariant under complex conjugation on X (C). A principal divisor 2 is a compactified divisor of the form (div f, − log |fC| ), where f is a rational function on X . This is a well defined notion, as can be seen from the Poincar´e-Lelongequation ([GH78], p. 388)
ddc(− log |f |2) + δ = 0. C div fC Let Div(d X ) be the set of compactified divisors. It is an abelian group under the operation (D, g) + (D0, g0) = (D + D0, g + g0). From remark 2.13 it is easily verified that this operation is well defined. The neutral element is (0, 0), that is, the empty divisor together with the zero function. The set of principal divisors is a subgroup of the group of compactified divisors. 2 1 The L1-arithmetic Chow group CH[(X ) is defined as the quotient of the group of compactified divisors by the subgroup of principal divisors. 2 Let X = X (C). Let µ be a real-valued 2-current in L−1(X) such that Z (3.1) µ = 1. X A compactified divisor (D, g) is said to be admissible with respect to µ (or µ-admissible) if the following equality of currents holds:
c dd g + δD(C) = deg D(C) µ. 2 In this situation we will also say that g is a µ-admissible L1-Green function for D. 2 The L1-Green function g is determined by D up to a function which is constant on every connected component of X. Indeed, if (D, h) is µ-admissible, then g − h is an harmonic function on a compact Riemann surface. Hence, it must be constant on each connected component of X by the maximum modulus principle. We remark that (3.1) is a necessary condition for the existence of admissible compacti- fied divisors (D, g) with deg D(C) 6= 0. To give a sufficient condition we use an existence result coming from the theory of elliptic operators: 2 Lemma 3.1. Let µ ∈ L−1(X). The equation (3.2) ddcu = µ 2 R has a solution u ∈ L1(X) if and only if X µ = 0. If µ is real-valued then u can be taken to be real-valued. If we take µ to be a C∞ form (in which case u ∈ C∞(X)), this is a well known consequence of the Hodge theorem in differential geometry (cf. [War71], 6.8). Equation (3.2) can be solved locally via standard techniques from the theory of elliptic PDE’s, namely the Lax-Milgram theorem and the Fredholm alternative. To prove that it is possible to patch together these local solutions to get a global solution, we shall use the cohomology machinery. We explain the precise steps giving a proof of Lemma 3.1 in the Appendix. 16 2 R Proposition 3.2. Let µ be a real-valued 2-current in L−1(X) such that X µ = 1. Then 2 for every Weil divisor D on X , there exists a L1-Green function g such that (D, g) is µ-admissible.
Proof: let µ0 be the canonical Arakelov 2-form if the genus of X is positive and the Fubini-Study form if it is zero. Letg ˜ be a real-valued solution to the equation of currents ([Lan88], Theorem 1.4) c dd g˜ + δD(C) = deg D(C) µ0. 2 R Letµ ˜ = deg D(C) (µ0 − µ). Thenµ ˜ belongs to L−1(X) and X µ˜ = 0. Using Lemma 2 c 3.1, we have a real-valued u ∈ L1(X) such that dd u =µ ˜. Then g :=g ˜ − u is a µ- 2 admissible, L1-Green function for D(C).
Principal divisors are admissible with respect to any 2-current. The set of µ-admissible 1 compactified divisors forms a group. We denote by CH[(X )µ the quotient of this group 1 1 by the subgroup of principal divisors. The natural map CH[(X )µ → CH[(X ) is an embedding of abelian groups. 1 Let x1, x2 ∈ CH[(X ) be represented by the compactified divisors (D1, g1) and (D2, g2). We suppose that the representatives have been chosen without common irreducible com- ponents. The intersection pairing is defined as
gi = hi + ψi c as in (2.11), and we define wi = dd hi + δDi(C). Then,
Z Z Z Z 1 1 ¯ (3.3)
where h1 ∗ h2 = h1w2 + h2δD1(C). This number does not depend of the choice of hi, ψi.
Example 3.3. For every compactified divisor Dˆ = (D, g) and c ∈ R, we have that c <(0, c), D>ˆ = [K : ] deg D(K). 2 Q Let ϕ : X → Y be a generically finite morphism between arithmetic surfaces. If we have a regular splitting g = h + ψ, then by definition ϕ∗g = ϕ∗h + ϕ∗ψ. As ϕ∗h is not a Green function with C∞ regularity when ϕ is ramified, this is not a regular splitting of ϕ∗g. However, we show in the following lemma that this decomposition can still be used to compute intersections.
Lemma 3.4. Let (Dx, gx) (resp. (Dy, gy)) be a compactified divisor on X (resp.Y). We decompose gx = hx + ψx (resp. gy = hy + ψy) as in (2.11) and we put w(ϕ∗hx) := c c dd ϕ∗hx + δϕ∗Dx(C) (resp. w(hy) := dd hy + δDy(C)). we have that Z Z Z Z 1 1 ¯ <ϕ∗gx, gy> = ϕ∗hx ∗hy + ϕ∗ψxw(hy)+ ψyw(ϕ∗hx)+ ∂(ϕ∗ψx)∧∂ψy . 2 Y Y Y 2πi Y Moreover, all of the integrals above are absolutely convergent. 17 Proof: Let Y = Y(C). Using Theorem 2.16, we obtain an auxiliary regular splitting
(3.4) ϕ∗gx = h + ψ, ∞ 2 where h is a Green function for ϕ∗Dx with C regularity, and ψ ∈ L1(Y ). Hence, we have that
Z Z Z Z 1 1 ¯ <ϕ∗gx, gy> = h ∗ hy + ψw(hy) + ψyw(h) + ∂ψ ∧ ∂ψy . 2 Y Y Y 2πi Y We deduce from (3.4) that
Z Z Z (3.5) ϕ∗hx ∗ hy = h ∗ hy + (ψ − ϕ∗ψx)w(hy). Y Y Y ∞ 2 As w(hy) is C and ψ − ϕ∗ψx is L , this equality shows the absolute convergence of the integral on the left. Thus, we find that Z Z ψyw(ϕ∗hx) = ψyw(h + ψ − ϕ∗ψx)(3.6) Y Y Z Z c = ψyw(h) + ψydd (ψ − ϕ∗ψx) Y Y The last integral is absolutely convergent because Z Z c 1 ψydd (ψ − ϕ∗ψx) = ∂ψy ∧ ∂(ψ − ϕ∗ψx), Y 2πi Y 2 and the functions on the right are L1. This shows that the left hand side of (3.6) is absolutely convergent. From this equality we deduce that Z Z Z Z 1 ¯ 1 ¯ ψyw(ϕ∗hx) + ∂(ϕ∗ψx) ∧ ∂ψy = ψyw(h) + ∂ψ ∧ ∂ψy. Y 2πi Y Y 2πi Y Together with (3.5), this proves both the absolute convergence of the first integral on the left and the desired formula.
3.2. Functoriality with respect to pull-back and push-forward. Let ϕ : X → Y be a generically finite morphism between arithmetic surfaces. Let (D, gD) (resp. (E, gE)) be a compactified divisor on X (resp. on Y). We define
∗ ∗ ∗ ϕ (E, gE) := (ϕ E, ϕ gE)
ϕ∗(D, gD) := (ϕ∗D, ϕ∗gD). Theorem 3.5. (1) The formulas above induce homomorphisms
ϕ∗ : CH[1(Y) −→ CH[1(X )
1 1 ϕ∗ : CH[(X ) −→ CH[(Y). (2) Let µ ∈ L2 X ( ) and ν ∈ L2 Y( ) be real-valued and such that R µ = −1 C −1 C X (C) R ν = 1. The formulas above induce homomorphisms Y(C) 18 ∗ 1 1 ϕ : CH[(Y)ν −→ CH[(X ) ϕ∗ν deg ϕ 1 1 ϕ∗ : CH[(X )µ −→ CH[(Y)ϕ∗µ. (3) Projection formula: for x ∈ CH[1(X ) and y ∈ CH[1(Y), we have that ∗ <ϕ y, x> =
ˆ ∗ ∗ T (D, g) := (p∗q D, p∗q g)(3.7) ∗ T µ := p∗q µ The following statement is a straightforward consequence of Theorem 3.5 and the definitions above. Corollary 3.7. Let T be a correspondence on the arithmetic surface X . (1) The formula (3.7) induces a homomorphism
Tˆ : CH[1(X ) −→ CH[1(X ). (2) If T is symmetric, then the morphism Tˆ is self-adjoint, i.e. for all x, y ∈ CH[1(X ),
ˆ 1 1 T : CH[(X )µ −→ CH[(X ) T µ . deg q Corollary 3.8. Let S be another correspondence on X . If S and T commute as endo- morphisms of Div(X ), then Sˆ and Tˆ commute as endomorphisms of Divd(X ). Proof: let (D, g) be a compactified divisor. Let g = h+ψ be a decomposition as in (2.11). ∞ 2 Let ψn be a sequence of C functions converging to ψ in L1 X (C) . The hypothesis implies that S and T commute as correspondences on X (C), so ST ψn − T Sψn ≡ 0. By Proposition 2.4, we have that ST ψ = T Sψ. By Theorem 2.16, we deduce that 2 ST g − T Sg = ST h − T Sh ∈ L1 X (C) . The function h is C∞ outside of the finite set F := {x such that |ST [x]|∩|D(C)|= 6 ∅}. For any point x ∈ X (C) − F , we have ST h(x) = T Sh(x) by hypothesis, so ST h = T Sh 2 in L1 X (C) . This shows that ST (D, g) = TS(D, g), as required.
19 Proof of Theorem 3.5: To prove (1), it suffices by Theorem 2.16 to show that principal divisors are sent into principal divisors. Let f be a rational function on X . Let Nϕ : K(X ) → K(Y) be the norm map between function fields induced by ϕ. We have that
2 2 ϕ∗(div f, − log |f| ) = (div Nϕ(f), − log |Nϕ(f)| ). On the other hand, if f is a rational function on Y, we have that ϕ∗(div f, − log |f|2) = (div ϕ∗f, − log |ϕ∗f|2). This finishes the proof of (1). Now we prove the first statement of (2). Let (E, g) be a compactified divisor on Y which is admissible with respect to ν. Theorem 2.16 shows that ϕ∗(E, g) is a compactified divisor on X . We need to check the equality of currents ∗ c ∗ ∗ ϕ ν dd (ϕ g) + δ ∗ = deg ϕ E( ) . ϕ E(C) C deg ϕ Let φ ∈ C∞(X). Using Lemma 2.17, we have that
c ∗ c dd (ϕ g)[φ] = dd g[ϕ∗φ] = −δE(C)[ϕ∗φ] + deg E(C) ν[ϕ∗φ] = −δϕ∗E(C)[φ] + deg E(C) ϕ∗φ · ν. Using Lemma 2.6 and the fact that deg ϕ∗E(C) = deg ϕ deg E(C), this finishes the proof of the statement concerning ϕ∗ in (2). To prove the part concerning ϕ∗ in (2), let (D, g) be a compactified divisor on X which is admissible with respect to µ. Theorem 2.16 shows that ϕ∗(D, g) is a compactified divisor on Y. As deg ϕ∗D(C) = deg D(C) , we need to check the equality of currents c dd (ϕ∗g) + δϕ∗D(C) = deg D(C) µ. Let φ be a C∞ function on Y . Using Lemma 2.17, we have that
c c ∗ dd (ϕ∗g)[φ] = dd g[ϕ φ] ∗ ∗ = −δD(C)[ϕ φ] + deg D(C) µ[ϕ φ] ∗ = −δϕ∗D(C)[φ] + deg D(C) ϕ φ · µ. As before, we conclude by Lemma 2.6. This finishes the proof of (2). Now, we prove (3). Let (Dx, gx) (resp. (Dy, gy)) be a member in the class x (resp. y), ∗ chosen such that |Dx(C)| ∩ |ϕ Dy(C)| = ∅. We decompose
gx = hx + ψx, gy = hy + ψy,
∞ ∗ as in (2.11). As w(hx) is a C 2-form, there exists a neighborhood U of |ϕ Dy(C)| and ∞ c s ∈ C (U) with w(hx) = dd s on U. We suppose that the support of ϕ∗hy is contained in U. Using Lemma 3.4, we have that Z Z Z Z ∗ ∗ ∗ ∗ 1 ∗ ¯ <ϕ gy, gx> = ϕ hy ∗ hx + ϕ ψyw(hx) + ψxw(ϕ hy) + ∂(ϕ ψy) ∧ ∂ψx. X X X 2πi X We will prove the required equality term by term. 20 Z Z ∗ ∗ ∗ ϕ hy ∗ hx = ϕ hyw(hx) + hx ϕ Dy(C) X X Z ∗ = ϕ hyw(hx) + ϕ∗hx Dy(C) U Z ∗ c = ϕ hydd s + ϕ∗hx Dy(C) U Z c (3.8) = hydd ϕ∗s + ϕ∗hx Dy(C) ϕ(U) Z = hyϕ∗w(hx) + ϕ∗hx Dy(C) ϕ(U) Z = hy ∗ ϕ∗hx, Y where we have used Lemma 2.19 in (3.8). Using Lemma 2.6, we obtain that Z Z ∗ ϕ ψyw(hx) = ψyϕ∗ w(hx) X Y Z = ψyw(ϕ∗hx). Y R ∗ R The equality X ψxw(ϕ hy) = Y ϕ∗ψxw(hy) is handled similarly. Using Lemma 2.5 we obtain that Z Z ∗ ¯ ¯ ∂(ϕ ψy) ∧ ∂ψx = ∂(ψy) ∧ ∂ϕ∗ψx. X Y
4. Hecke correspondences on modular curves
4.1. Integral models of the modular curves X0(N) and Hecke correspondences. 1 Let N ≥ 1 be an integer. We consider the modular curve X0(N) := Γ0(N)\ H ∪ P (Q) . This Riemann surface admits a smooth, projective model over Q. In order to obtain a model over Z, we will use the modular interpretation in terms of Γ0(N)-structures `ala Drinfeld as in [KM85]. We recall the basic facts and definitions: • Let S be a scheme and let E → S be an elliptic curve. A group subscheme G ⊂ E over S is cyclic of order N if it is locally free of rank N and there exists a morphism T → S which is faithfully flat and locally of finite presentation and a T -point P ∈ G(T ) such that we have the equality of Cartier divisors N X G ×S T = [aP ]. a=1 • An isogeny π : E → E0 over S is said to be cyclic of order N if ker π is a cyclic subgroup of order N in the above sense. • Let d be a divisor of N. There is a unique cyclic subgroup Gd ⊂ G, called the standard cyclic subgroup of order d, characterized by: if P is a f.p.p.f.3 local generator of G, then Gd is generated by (N/d)P (Theorem 6.7.2. of loc. cit.).
3Faithfully flat of finite presentation (Fid`elementplat, de pr´esentationfinie in french). 21 Let X0(N) be the compactified coarse moduli scheme associated to the moduli problem [Γ0(N)] classifying cyclic N-isogenies between elliptic curves as constructed in loc. cit. Ch. 8. Alternatively, the functor [Γ0(N)] can be described as classifying cyclic subgroups of order N of elliptic curves. The scheme X0(N) is proper and normal. It has good reduction at p - N and bad reduction at p|N. We consider the related functor
FN : Sch −→ Sets which attaches to a scheme S the set
FN (S) = {(E,C) such that E/S is an elliptic curve and C is a Γ0(N)−structure on E}/ ∼, where ∼ stands for the natural isomorphism notion. Let l be a prime number such that l - N. We define morphisms
αl, βl : FNl → FN as follows. Let S be a scheme, and let (E,C) represent a class in FNl(S). Let CN ⊂ C (resp. Cl ⊂ C) be the standard cyclic subgroup of order N (resp. l). We put
αl(E,C) := (E,CN ), βl(E,C) := (E/Cl, C/Cl).
The morphism βl is well defined because of Theorem 6.7.4. of loc. cit. Using the coarse moduli property and the construction of the compactification as a normalization, we obtain finite morphisms αl, βl : X0(Nl) → X0(N). We define the Hecke correspondence of order l by ∗ Tl := (βl)∗αl . Let d be a divisor of N such that (d, N/d) = 1. We define the Atkin-Lehner involution wd on X0(N) first as a functor FN → FN as follows: let S be a scheme and let (E,C) represent a class in FN (S). Let Cd be the standard cyclic subgroup of order d of C. We put 1 w (E,C) := E/C , (E[N] ∩ C)/C . d d d d As before, we obtain an involution wd : X0(N) → X0(N).
In what follows we will suppose N squarefree to simplify the analysis of the bad fibers. For a prime number p, we put Xp(M) := X0(M) ⊗ Fp. If p|N, then Xp(N) has two ∞ irreducible components, each one of them isomorphic to Xp(N/p). We denote by Xp (N) 0 N N (resp. Xp (N)) the irreducible component intersecting D∞ (resp. D0 ). Lemma 4.1. Suppose that N is squarefree. For prime numbers p|N, l - N and u ∈ {0, ∞}, we have that ∗ u ∗ u u αl Xp (N) = βl Xp (N) = Xp (lN), u u u (αl)∗Xp (lN) = (βl)∗Xp (lN) = (l + 1)Xp (N), ∞ 0 0 ∞ wN Xp (N) = Xp (N), wN Xp (N) = Xp (N). Nl Nl N N N Proof: We first remark that (αl)∗Du = (βl)∗Du = Du , and wN D∞ = D0 . (It is enough to check these equalities on the generic fiber, where it is clear). Using the notations 1/0 = ∞ and 1/∞ = 0, we have that
∞ N ∞ N
∗ ∞ Nl ∞ Nl <αl Xp (N),Du > =
∗ ∞ ∞ (4.1) αl Xp (N) = Xp (Nl). ∗ 0 0 A similar argument applies to prove αl Xp (N) = Xp (Nl). The analogous claims con- ∗ cerning βl are proved in the same way. Now, for the push-forward, we have that
∞ N ∞ ∗ N <(αl)∗Xp (Nl),D∞> =
Lemma 4.2. The correspondence Tl is symmetric. Proof: this is well known on the generic fiber. Hence, for any irreducible, horizontal divisor D on X0(N), we have that ∗ ∗ (βl)∗αl D = (αl)∗βl D + V, where V is a vertical divisor. (This is justified because both sides of this equality are equal on the generic fiber). As αl and βl are finite, no vertical divisor is produced during the pull-back operation. Hence, V = 0. On the other hand, if D is an irreducible com- ponent of a finite fiber, we conclude by Lemma 4.1.
N The cusps Γ0(N)∞ and Γ0(N)0 induce Q-points of X0(N). We denote by D∞ (resp. N D0 ) the corresponding Zariski closure in X0(N). N N Lemma 4.3. we have that TlD∞ = (l + 1)D∞. Proof: An explicit calculation shows that the equality holds on the generic fiber. Then N N TlD∞ −(l +1)D∞ must be a vertical divisor V . Since the morphisms αl, βl defining Tl are finite, no vertical component is produced during the pull-back operation. Hence, V = 0.
dz∧dz¯ 4.2. Arithmetic Chow group of X0(N). Let µ = 2 be the hyperbolic 2-form on Im(z) + H. This form is GL2 (R)-invariant, and induces a finite measure on X0(N). We denote by µN := µ/V olµ X0(N) the normalized 2-form on X0(N).
2 Lemma 4.4. The 2-form µN belongs to L−1 X0(N) . 23 Proof: It suffices to verify the assertion around the singularities of µN , namely cusps c dx∧dy µ1 and elliptic points. As dd log y = − 4πy2 = − 12 , we only need to show that log y defines 2 a L1 function on a neighborhood of these points. ∗ ∗ Cusps: Let s be the width of the given cusp, and let σ = ∈ GL ( ) be u v 2 R the element taking it to ∞. In terms of the local parameter q = e2πiσ(z)/s, we have the relation
s (4.2) y = −(log |q|) |j (z)|2, 2π σ 2 where jσ(z) = uz + v. From this, it is easily checked that log y is L1 in a neighborhood of q = 0. Elliptic points: Let z0 = x0 + iy0 be an elliptic point of order n. A local parameter is n τ = z−z0 . After the choice of a branch of logarithm, we have that z−z¯0
1 − |τ|2/n (4.3) y = y almost everywhere. 0 |1 − τ 1/n|2 2 From this formula it is easy to verify that log y is L1 in a neighborhood of τ = 0.
1 We introduce the notation CHd(N) := CH[ X0(N) . µN
Theorem 4.5. (1) The correspondences Tl with l - N and wd with d|N, d - N/d, ˆ 1 induce homomorphisms Tl and wˆd on CH[(X0(N)). (2) These homomorphisms preserve CHd(N). ˆ (3) The morphism Tl is self-adjoint. 1 ˆ (4) Suppose N squarefree. The sub-algebra of End CH[ X0(N) spanned by Tl with
l - N and wˆd with d|N is commutative. Proof: The assertion (1) is an immediate consequence of Theorem 3.5,(1). It is easily verified that TlµN = (l+1)µN . Then, using Lemma 4.4 and Theorem 3.5(2), we conclude (2). Assertion (3) follows from Corollary 3.7,(2) and Lemma 4.2. To prove (4), we first note that the sub-algebra of End Div X0(N) spanned by the correspondences Tl and
wd is commutative (cf. [Shi94], Proposition 3.32 to check commutativity for horizontal divisors and Lemma 4.1 for vertical divisors). Then we use Corollary 3.8 to deduce the claim.
N The next Lemma will be used to construct a µN - admissible Green function for D∞.
Lemma 4.6. (1) Let P∞ := Γ0(N)∞ ∈ X0(N). There exists a weight k ∈ 12N cuspidal form f ∈ Sk Γ0(N) and an integer r ∈ Z>0 such that
(4.4) div f = r[P∞] on X0(N).
Moreover, k/r = 12/[Γ0(1) : Γ0(N)]. (2) Suppose f1 ∈ Sk1 Γ0(N) , with k1 ≡ 0 mod 2, satisfies the equality div f1 = ∗ r1[P∞] for some positive integer r1. Then there exists A ∈ C such that
r1 r f = Af1 .
Moreover, k1/r1 = 12/[Γ0(1) : Γ0(N)]. 24 Proof: if N = 1, we consider the discriminant function
∞ Y n 24 2πiz ∆(z) = q (1 − q ) ∈ S12 Γ0(1) , q = e , n=1 satisfying div ∆ = [P∞]. If N > 1, we consider ∆ as a modular form in S12 Γ0(N) . We have that
X div (∆) = aC [C], C cusp P where aC > 0 for every cusp C and C cusp aC = [Γ0(1) : Γ0(N)] =: dN . The divisor
div (∆) − dN [P∞] has degree 0 and is supported on the cusps. By the Manin-Drinfeld theorem ([Dri73], [Elk90]), there exists a meromorphic function g on X0(N) and a positive integer n such that
div (g) = n(div (∆) − dN [P∞]). n Thus, f := ∆ /g is an holomorphic modular form in S12n Γ0(N) that satisfies (4.4). We have that
div (f) = ndiv ∆ − div g = ndN [P∞],
so r = n[Γ0(1) : Γ0(N)] and k/r = 12/[Γ0(1) : Γ0(N)], which proves the first assertion. r r1 To prove the second, we remark that since the holomorphic modular form f1 /f has no zeros, it must be constant. We obtain that k1/r1 = k/r = 12/[Γ0(1) : Γ0(N)], i.e. by writing down that its weight is 0.
Let f be as in part (1) of the preceding Lemma. We define
1 (4.5) g (z) := − log |f(z)2yk|. ∞ r
The statement (2) in the Lemma implies that g∞ depends on the choice of f only up to an additive constant. An alternative proof of this fact is given by part (1) of the following proposition:
2 Proposition 4.7. (1) g∞ induces a µN -admissible L1-Green function for [P∞]. 1 (2) For c ∈ P (Q), put Q = Γ0(N)c. We have that
2 g∞(z) = −δP∞,Q log |q| − bN log(− log(|q|)) + b, z → c, ∞ where q is the standard local chart around Q, the function b is C and bN = 12/[Γ0(1) : Γ0(N)].
Proof: let z0 = x0 + iy0 ∈ H be a point of order n and let τ be the local chart around z0 as in (4.3). If n = 1 (i.e. if z0 is not an elliptic point), the expression (4.3), shows that ∞ 2 g∞|Y0(N) is C at z0. If n > 1, g∞ is continuous at z0 and belongs to L1. We have that 25 k ddcg (z) = − ddc log y ∞ r 12 µ = − − 1 [Γ0(1) : Γ0(N)] 12 = µN . The q-expansion of f at Q is given by
∞ k X n f|kσ(z) = jσ(z) f(z) = anq , arQ 6= 0, n=rQ
where rQ is the order of vanishing of f at Q. Using the relation (4.2), we obtain that
yk|f(z)|2 = (− log |q|)k|q|2rQ |h(q)|2, where h is an holomorphic function defined on a disc containing 0 and such that h(0) 6= 0. We find that
2 k 1 2 gP (z) = −δP,Q log |q| − log(− log |q|) − log |h(q)| rQ rQ and bN = k/rQ by Lemma 4.6,(1).
N ˆ Let D∞ := D∞. We denote by D∞ = (D∞, g∞) the compactified divisor obtained by the proposition above. Lemma 4.8. We have that ˆ ˆ ˆ TlD∞ = (l + 1)D∞ + (0, cN,l), 12(l−1) with cN,l = log(l). [Γ0(1):Γ0(N)]
Proof: as TlD∞ = (l + 1)D∞ (cf. Lemma 4.3), the difference Tlg∞ − (l + 1)g∞ is a constant c. To evaluate c, we compare the expansion around ∞ of both functions. With the notations of equation (4.5), we consider the expansion at ∞ of f:
r r+1 f(z) = arq + ar+1q + . . . , ar 6= 0, r ≥ 1. 2 r 2 We write |f(z)| = |arq | h(q), where h is defined on a neighborhood of 0 and h(0) = 1. On the other hand, if we put
1 j if 0 ≤ j ≤ l − 1 0 l (4.6) αj := l 0 if j = l. 0 1 then we have that
l 1 X k (4.7) T g (z) = − log | Im(α z) f(α z)2|. l ∞ r j j j=0 But, 26 1 − 2πl log(|q|) si 0 ≤ j ≤ l − 1 Im(αjz) = l − 2π log |q| si j = l and
r/l |arq |hj(q) if 0 ≤ j ≤ l − 1 |f(αjz)| = rl |arq |hl(q) if j = l,
where the functions h0, . . . , hl are defined around 0, positive, continuous and satisfy hj(0) = 1. Using these notations, we obtain that
l k 1 X 1 T g (z) − (l + 1)g (z) = (l − 1) log l − log |h (q)|2 + log |h(q)|2. l ∞ ∞ r r j r j=0 Hence, taking the limit as z → ∞ and using Lemma 4.6,(2) we obtain that c = l−1 bN log(l ).
Definitions 4.9. • For a divisor D ∈Div(X0(N)Q), we denote by D its Zariski closure in X0(N). If deg D = 0, we denote by Φ(D) a vertical divisor (which may have rational coefficients) such that D+Φ(D) has degree zero on every irreducible component of every vertical fiber of X0(N). The divisor Φ(D) is well defined up to a sum of rational multiples of finite fibers. 0 • Suppose that the genus of X0(N) is nonzero. We identify J0(N)(Q) with P ic (X0(N)Q) using the rational point Γ0(N)∞. Let
i∞ : J0(N)(Q) → CHd(N) 2 be given by i∞(D) = (D + Φ(D), gD), where gD is a µN -admissible L1-Green function for D normalized by the condition ˆ
As deg D = 0, the function gD does not depend on the choice of a normalization for g∞ (cf. example 3.3).
Lemma 4.10. The function i∞ is well defined. Moreover, it is an embedding of abelian groups.
Proof: We must check that i∞(D) does not depend of the choice of Φ(D). Let Xp := 2 X0(N) ⊗ Fp be a finite fiber. Let gp be a µN -admissible L1-Green function for D such that ˆ <(D + Φ(D) + Xp, gp), D∞> = 0.
As (Xp, gp −gD) = (D+Φ(D)+Xp, , gp)−i∞(D) is a compactified divisor, the difference gp − gD is a constant c. Moreover, using the example 3.3, the equality ˆ <(Xp, c), D∞> = 0 unwinds to c log p + = 0. 2 2 This gives (Xp, gp − gD) = (div p, − log p ), that is a (compactified) principal divisor. The injectivity of i∞ then follows plainly from the definitions.
27 Corollary 4.11. (1) The following diagram commutes:
Tl J0(N)(Q) / J0(N)(Q) _ _
i∞ i∞ ˆ Tl CHd(N) / CHd(N).
(2) Let <, >NT denote the N´eron-Tate height pairing on J0(N)(Q). We have that
(4.8) TlgD = gTlD. 2 Theorem 4.5,(2) ensures that TlgD is an admissible L1-Green function for TlD. Hence, (4.8) holds up to a constant. The constant must be zero since by Theorem 4.5(3) and Lemma 4.8, we have that
ˆ ˆ ˆ <(TlD,TlgD), D∞> = <(D, gD), TlD∞> ˆ = (l + 1)<(D, gD), D∞> + <(D, gD), (0, cN,l)> = 0, i.e., by the definition of gD and the fact that deg D = 0 (cf. Example 3.3).
Theorem 4.5,(3) and the previous Corollary give the following result:
Corollary 4.12. The Hecke correspondence Tl : J0(N)(Q) → J0(N)(Q), with l - N, is self-adjoint with respect to the N´eron-Tate height pairing. Remark 4.13. This result is not new (cf. [GZ86]). It can be proved using the fact that J0(N)(Q) ⊗ C and S2(Γ0(N)) are isomorphic as Hecke modules. The latter algebra is diagonalizable, which implies that the former is also diagonalizable, which in turn implies the self-adjointness.
4.3. Arithmetic Chow group of X0(N) with real coefficients. In this Section, N is a fixed squarefree integer. Let Divd R(N) denote the R-vector space made of pairs (D, g), 2 with D a Weil divisor with real coefficients on X0(N) and g a µN -admissible L1-Green function for D(C). Let PR(N) be the sub-espace spanned by the principal divisors. We put
CHd R(N) := Divd R(N)/PR(N). Definition 4.14. Let
KN = {x ∈ CHd R(N)|
num CHd R (N) := CHd R(N)/KN 28 Remarks 4.15. • The arithmetic intersection pairing <, > extends to a nondegen- num num erate bilinear form on CHd (N) × CHd (N). num R R • Let σ : CHd(N) → CHd R (N) be the natural map. We have an exact sequence
i∞ σ num 0 / J0(N)(Q)tors / CHd(N) / CHd R (N).
Indeed, if <σ(x), y> = 0 for every y ∈ CHd R(N), then by varying y over all classes of compactified divisors of the form (F, 0) and (0, c) with F an irreducible component of a fiber of X0(N) and c ∈ R we conclude that the underlying divisor of any representative of x comes from a divisorx ˜ ∈ J0(N)(Q). Then the N´eron- Tate height ofx ˜ vanishes because of the Faltings-Hriljac formula, implying that it is a torsion element.
ˆ u u ˆ ˆ ∞ ˆ 0 For u ∈ {0, ∞} and p|N, we put Xp := (Xp (N), 0). Let Gp := Xp − Xp and let F := {(0, c), c ∈ R}. We define
ˆ ⊥ ˆ num Eis := (F ⊕ RD∞) ⊕ ⊕p|N RGp ⊂ CHd R (N). ⊥ Here, the notation A ⊕ B means that A ⊕ B is a direct sum of vector subspaces and that ˆ A and B are mutually orthogonal. We remark that the space F ⊕ RD∞ does not depend on the normalization of g∞. The Hodge index theorem in this context can be written as Theorem 4.16. We have that
num ⊥ CHd R (N) = Eis ⊕ J, where J is, by definition, the orthogonal complement of Eis. Moreover, the rule (D, g) 7→ D ∼ induces an isomorphism of Hecke modules J = J0(N)(Q) ⊗ R. The proof of the Hodge index theorem given in [MB85], p. 85, works in this situation ˆ (cf. also [Bos99], Theorem 5.5). We just remark that the choice of basis Gp is convenient because it forces an element in the orthogonal complement of Eis to have degree 0 on every irreducible component of every finite fiber. The compatibility of the Hecke actions on J and J0(N)(Q) ⊗ R is given by Corollary 4.11(1). The following assertion is a simple consequence of Lemma 4.1 and Lemma 4.8. ˆ ˆ Proposition 4.17. (1) The spaces F and RGp with p|N are eigenspaces for Tl and ˆ ˆ wˆd with d|N. More precisely, Tlx = (l + 1)x for all x ∈ F ⊕ Gp and the morphism ˆ wˆd is the identity on F . On Gp, wˆd is the identity (resp. −wˆd is the identity) if w (X∞) = X∞ (resp. if w (X∞) = X0). In particular, w | = −id . d p p d p p N Gˆp Gˆp ˆ ˆ (2) The space F ⊕ RD∞ is stable under Tl. More precisely,
ˆ ˆ ˆ ˆ Tl(0, c) = (l + 1)(0, c), TlD∞ = (l + 1)D∞ + (0, cN,l), 12(l−1) with cN,l = log(l). [Γ0(1):Γ0(N)] 29 4.4. Self-intersection and refined self-intersection of the dualising sheaf. Let ω 1 be the dualising sheaf on X0(N). This sheaf induces a class in CH (X0(N)), and we also denote by ω a divisor in this class. Then ωQ := ω ⊗ Q is a canonical divisor on X0(N)Q. √Let π : X0(N)Q → X0(1)Q be the natural morphism and let i (resp. j) be the orbit of iπ/3 −1 (resp. e ) in X0(1). Using Hurwitz’s formula and the Manin-Drinfeld theorem ([Elk90], [Dri73]) we find that
1 (4.9) ωQ = (2g − 2)[∞] − Hi − 2Hj in CH (X0(N)Q) ⊗ Q.
Here, g is the genus of X0(N), and the divisor Hi (resp. Hj) is the sum of all divisors 1 1 of the form 2 [P ] − [∞] (resp. 3 [P ] − [∞] ) with π(P ) = i (resp. π(P ) = j) and π unramified at P . Equivalently, the points P appearing in Hi (resp. Hj) are the Heegner points of discriminant -4 (resp. -3) on X0(N) (cf. [MU98], Section 6). We introduce the following notation. For a divisor E ∈Div(X0(N)Q), we write DE for its Zariski closure in X0(N). We deduce from (4.9) the equality
1 (4.10) ω = (2g − 2)D∞ − DHi − 2DHj + V in CH X0(N) ⊗ Q, where V is a vertical divisor contained inside the space of fibers of bad reduction. This comes from the fact that both sides of (4.10) are equal on the generic fiber. Let W be a vertical divisor such that ωJ := ω − (2g − 2)D∞ + W has degree zero on each irreducible component of every fiber of X0(N). The divisor ωJ is then identified with a point in J0(N)(Q). We defineω ˆJ := i∞(ωJ ) ∈ J (whereω ˆJ does not depend on the choice of W ) and
ωEis := ω − ωJ . ˆ We will now specify a compactification of ωEis. We define W = (W, c), with c a constant ˆ ˆ ˆ such that
2 2 12 (4.11) kfC(z)kP et = |fC(z)| (4πy) , z = x + iy ∈ H. This factor 4π in (4.11) is motivated by a natural isomorphism between the line bundle of weight 12 modular forms attached to Γ(N) and the 12th power of the line bundle on X(N) induced by the canonical bundle on the universal elliptic curve. This last line 2 bundle has a canonical metric (the so-called L metric) and k·kP et in this case is obtained by following the isomorphism (see [K¨uh01], Section 4.14). We denote by ∆ the section of M12(Γ0(N)) defined by the discriminant function. Using Proposition 4.7, we see that the pair ˆ 2 E := (div (∆), − log k∆kP et) 30 is a compactified divisor. The product expansion of the discriminant function ∆ shows that 2 12 log(2) a(− log k∆kP et) = − . [Γ0(1) : Γ0(N)]
This data defines an elementω ˆEis ∈ Eis. Finally, we have the elementω ˆ =ω ˆEis +ω ˆJ ∈ num 2 CHd R (N). The main ingredient needed for the computation ofω ˆEis is a calculation due to U. K¨uhn.We deal mostly with the intersections at the bad fibers. Let us postpone the proof of Theorem 1.4 to the following Section. If there are no elliptic points on X0(N) (i.e. if there are two different prime divisors p, q|N such that p, q∈ / {2, 3}, p ≡ 3 mod 4 and q ≡ 2 mod 3), then there are no nonra- mified points above the elliptic points of X0(1) ([Shi94], Proposition 1.43). In this case, ωˆ = 0 because the divisors D 0 et D 0 in (4.10) are empty. J Hi Hρ Theorem 4.16 allows to associate to a normalized eigenform f ∈ S2 Γ0(N) the f- isotypical component ofω ˆJ , which we denote byω ˆf . The Gross-Zagier formula and the Gross-Kohnen-Zagier theorem then allow us to compute the self-intersection of this ele- ment in some cases. √ Proof of Theorem 1.5: since both Q(i) and Q( −3) have class number one, the Heegner points in Hi and Hj are defined over these respective quadratic fields. Moreover, since Gal(Q(i)/Q) preserves the set of Heegner points of discriminant −4, the divisor Hi is defined over Q. The same argument applies to show that Hj is also defined over Q. ∗ ∗ Let X be the quotient of X0(N) by the Fricke involution wN , and let J be the ∗ ∗ jacobian of X . Let κ : X0(N) → X be the canonical map, which is of degree two. Let ωf := κ∗(ωf ). We remark that we have the equalities
(4.12) hNT (Hi)f = 2hNT (Hi)f , hNT (Hj)f = 2hNT (Hj)f , hNT ωf = 2hNT ωf . Indeed, we have that
∗ 2hNT ωf = <ωf , κ∗κ ωf > ∗ ∗ = <κ ωf , κ ωf > = hNT ωf + (wN )∗ωf = hNT 2ωf = 4hNT ωf , where the last equality comes from the fact that the Heegner points of given discriminant are permuted under wN . The same argument applies to check the other two equalities in (4.12). We then have that
2 − 2ˆωf = 2hNT ωf = hNT ωf = hNT (Hi)f + 4hNT (Hj)f + 4<(Hi)f , (Hj)f >NT It follows from the Gross-Kohnen-Zagier theorem [GKZ87] that the images in 31 J0(N)(Q)/wN f ⊗ R of the divisors Hi,Hj are collinear (see ch. V of loc. cit. for the case of even discriminant). In particular, 1/2 1/2 <(Hi)f , (Hj)f >NT = hNT (Hi)f hNT (Hj)f . This, together with the equalities (4.12), prove (1.8). Suppose that N is prime. Then Hi is either the empty divisor, or a divisor that contains two Heegner points that are interchanged by wN . In this situation, the calculation of hNT (Hi)f in terms of special values is given by [BY09], Corollary 7.8. The same argument applies to Hj and the calculation of hNT (Hj)f .
Remarks 4.18. • If N = p1p2 . . . ps is a squarefree integer with s > 1, then (Hj)f is either empty or contains 2s (f-isotypical components of) Heegner points. In the latter case, they give raise to 2s−1 (not necessarily different) elements of √ J0(N) Q( −3) f /wN ⊗ R. The Gross-Kohnen-Zagier theorem and the Gross- Zagier formula ensure that they are all collinear, and that they all have the same height. However, we do not know how to exclude the possibility that any given two of these points have opposite signs. This prevents us from computing hNT (Hj)f . In the case that s = 1, the technical improvements on the Gross-Zagier formula by Bruinier and Yang rule out this problem. • We also point out that the original Gross-Zagier formula as stated in [GZ86] does not apply to Heegner points of even discriminant, yet their methods undoubtedly apply in the case needed here, namely discriminant -4. On the other hand, the restriction on the parity has been eliminated in recent works (cf. [BY09], [Zha04], [Con04]).
2 4.4.1. Computation of ωˆEis. This Section is devoted to the proof of Theorem 1.4. As 2 2 ˆ 2 ˆ 2 ωˆEis = (2g − 2) D∞ + W , it suffices to prove the following two lemmas: Lemma 4.19. (U. K¨uhn,[K¨uh01]) We have that ˆ 2 144 1 0 D∞ = ζ(−1) + ζ (−1) . [Γ0(1) : Γ0(N)] 2 Proof: It is shown in [K¨uh01], Corollary 6.2, that the generalized self-intersection number 2 M is given by
2 1 M = 144[Γ (1) : Γ (N)] ζ(−1) + ζ0(−1) . 0 0 2 2 By [K¨uh01] Proposition 7.4, we have that Eˆ2 = M . The Manin-Drinfeld Theorem ([Dri73], [Elk90]) implies that
1 div ∆ = [Γ0(1) : Γ0(N)]D∞ + V in CH (X0(N)) ⊗ Q, where V is a vertical divisor. Thus, ˆ ˆ num [Γ0(1) : Γ0(N)]D∞ = E + (0, c) in CHd R (N). 2 As a(g∞) = a(− log k∆kP et), by definition, we obtain that c = 0. We conclude that
ˆ 2 −2 ˆ2 D∞ = [Γ0(1) : Γ0(N)] E , as required to conclude the proof.
32 Lemma 4.20. We have that X log p Wˆ 2 = −(g − 1)2 . g − 2gp + 1 p|N We break the proof of this lemma in two further lemmas.
Lemma 4.21. For p - N, we have that
∞ 0
∞ 0 (4.13)
ˆ ∞ ∞ 0 2 0 =
∞ ∞ 2 (2gp − 2) log p = <ω, Xp > + (Xp ) ∞ ∞ 2 = <ωEis,Xp > + (Xp ) ∞ ∞ ∞ 2 = (2g − 2)
∞ ∞ 0 (4.15)
∞ 0 2 0 = <(Wp, 0), (Xp + Xp , − log p )> ∞ 0 =
33 Lemma 4.22. For some c ∈ R, we have the equality
ˆ (g − 1) X −1 ˆ num W = − (g − 2gp + 1) Gp + (0, c) in CHd (N). 2 R p|N ˆ P ˆ num Proof: : We write W = p|N apGp + (0, c) in CHd R (N). Using Lemma 4.21, we find that
∞ (g − 1) log p =
= −2ap(g − 2gp + 1) log p, where we have used (4.13) in the last equality.
Proof of Lemma 4.20: using Lemma 4.22, we have that
g − 12 X Wˆ 2 = (g − 2g + 1)−2Gˆ2 2 p p p|N Using equation (4.13), we obtain that ˆ2 ∞ 2 0 2 ∞ 0 Gp = (Xp ) + (Xp ) − 2
= −4(g − 2gp + 1) log p, which concludes the proof.
2 5. Appendix: cohomology of L1 spaces on compact Riemann surfaces The goal of this Section is to prove Lemma 3.1 used in Section 3.1 to prove the existence 2 of admissible L1-Green functions (Proposition 3.2). The strategy to solve equation (3.2) is to first solve it locally, then to patch together the solutions to obtain a global one using cohomological machinery. The key local input to make this machinery work is the following
2 Theorem 5.1. Let U ⊂ C be a connected open set. Then for every g, f1, f2 ∈ L (U) the equation ∂2 ∂2 ∂f ∂f + u = g + 1 + 2 ∂x2 ∂y2 ∂x ∂y 2 has a solution u ∈ L1(U). This result is well known and can be handled by standard methods of PDE’s (e.g. the Lax-Milgram Theorem and the Fredholm alternative). A proof of this theorem can be found in [GT77], p. 170, Theorem 8.3. We will also need an appropriate version of the ∂¯ lemma: 34 Lemma 5.2. Let U ⊂ C be a connected open set, and let f ∈ L2(U). There exists a 2 u ∈ L1(U) such that ∂u¯ = fdz.¯
2 Proof: Theorem 5.1 implies that there exists a u0 ∈ L1(U) such that ∂∂u0 = ∂f ∧ dz¯. ¯ Let h := ∂u0 − fdz¯. As ∂h = 0, we have that h is harmonic. By the ellipticity of ∂∂, we deduce that h is C∞ (and in fact anti holomorphic because it is annihilated by ∂). Then, by a special case of the Dolbeaut lemma (cf. [For81], Theorem 13.2) there exists ∞ 2 a u1 ∈ C (U) such that ∂u1 = h. Then u := u0 − u1 belongs to L1(U) and verifies ∂u = fdz.¯ Let X be a compact Riemann surface. We will use the following property of the cohomology on X: Theorem 5.3. Consider the following exact sequence of sheafs on X: 0 → A → B →α C → 0. If H1(X,B) = 0, then H1(X,A) ∼= C(X)/α B(X). A proof of this theorem can be found in [For81], Theorem 15.13.
Let L(0,1)(X) (resp. L(1,0)(X)) be the space of 1-currents on X of type (0, 1) (resp. (1, 0)) which are locally L2. More precisely, an element ω ∈ L(0,1)(X) can be written on sufficiently small open sets U as fdz¯ with f ∈ L2(U) (and similarly for elements in L(1,0)(X)). Out of this definition we construct sheafs L(0,1) and L(1,0) in the obvious way. 2 ¯ 2 (0,1) If we consider L1 as a sheaf on X we have well defined morphisms ∂ : L1 → L and 2 (1,0) ∂ : L1 → L . Lemma 5.4. We have that 1 2 • H (X,L1) = 0 • H1(X, L(1,0)) = 0.
2 (1,0) ∞ As the sheafs L1 and L are stable under multiplication by C functions, this lemma can be proved carrying over the argument of [For81], Theorem 12.6, word by word with 2 (1,0) either of the sheafs L1 or L . Lemma 5.5. We have that (0,1) ¯ 2 L (X) = ∂L1(X) ⊕ Ω(X), where Ω is the sheaf of anti holomorphic 1-forms on X. Proof: The space L(0,1)(X) is endowed with the inner product Z (ω, η) = −i ω ∧ η. X 2 For f ∈ L1(X) and ω ∈ Ω(X), we have that Z (ω, ∂f) = −i d(ωf) = 0. X ¯ 2 This shows that the subspaces ∂L1(X), Ω(X) are orthogonal. Hence, it suffices to show that the codimension of the former equals the dimension of Ω(X), that is the genus of X. Let O be the sheaf of holomorphic functions on X. Consider the sequence 35 2 ∂¯ (0,1) 0 → O → L1 → L → 0. We claim that this sequence is exact. To check exactness in the middle, let U ⊂ X be 2 ¯ an open set, and let f ∈ L1(U) be such that ∂f = 0. This implies that f is harmonic, and so it must be C∞. But then, f must be holomorphic because it is annihilated by 2 ∂¯ (0,1) ∂. Since the surjectivity of L1 → L needs to be checked only locally, this map is surjective by Lemma 5.2. Using Lemma 5.4 and Theorem 5.3 we conclude that 1 ∼ (0,1) 2 H (X,O) = L (X)/∂L1(X). 1 As the dimension of H (X,O) equals the genus of X, this finishes the proof.
Proof of Lemma 3.1: the vanishing of the integral of µ is necessary, as can be checked using Lemma 2.2. To prove the reciprocal, we claim 1,0 0,1 2 (1) dL (X) = dL (X) = ∂∂L1(X) (2) we have an exact sequence (1,0) d 2 0 → Ω → L → L−1 → 0. Claim (1) follows from Lemma 5.5. To check exactness in the middle in claim (2), 2 ∞ we just remark that ∂∂f = 0 and that f ∈ L1(U) implies f ∈ C (U) because of the ellipticity of ∂∂. Then, ∂f is holomorphic because it is annihilated by ∂. Exactness on the right follows from Theorem 5.1. Using the claims (1) and (2), Lemma 5.4 and Theorem 5.3, we have that 1 ∼ 2 (1,0) 2 2 H (X, Ω) = L−1(X)/d L (X) = L−1(X)/∂∂L1(X). As the space H1(X, Ω) is 1-dimensional (a consequence of Serre duality, cf. [For81], 2 2 Theorem 17.11), the image of µ in L−1(X)/∂∂L1(X) must vanish (since otherwise all 2 elements in L−1(X) would have integral equal to zero). This proves the existence of a solution because ddc = −(2πi)−1∂∂. 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Facultad de Matematicas,´ Pontificia Universidad Catolica´ de Chile, Vicuna Mackenna 4860, Santiago, Chile E-mail address: [email protected]
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